MA6566 Discrete Mathematics, Third year, Departmentof Computer Science Engineering and Information Technology, First and secondUnits Questions

Part –A ( 10 x 2 = 20 )
1. Write the symbolic representation of “ if it rains today, then I buy an umbrella”.
2. Prove that if 3n+2 is odd then n is odd.
3. Define universal and existential quantifiers.
4. Find the truth table for p
5. Express A in terms of the connectives {
6. How many bit strings of length 10 contain i) exactly four 1’s     ii) atleast four 1’s?
7. Define permutation.
8. How many permutations are there in the word MISSISSIPPI?
9. State Pigeonhole principle.
10. Prove that .

Part-B ( 5 x 16 = 80 )

11.a) Show that R can be derived from the premises P ,  and Q.
b) Show that the hypothesis, “It is not sunny this afternoon and it is colder than yesterday,” “We will go swimming only if it is sunny,” “If we do not go swimming, then we will take a canoe trip,” and “If we take a canoe trip, then we will be home by sunset” lead to the conclusion “We will be home by sunset”.
12.a) Find the PDNF and PCNF of the formula P ∨ (¬P→(Q∨ (¬Q→R)))
b)Prove that (x)(P(x) by the method of contradiction.
13.a) Show that .
b) From a club consisting of six men and seven women , in how many ways we select a committee of
 i)3 men and 4 women                   ii) 4 person which has atleast one women            iii) 4 person that has at most one man                    iv) 4 persons that has children of both sexes.
14.a) Using generating function solve ,n given
b)Find the number of integers between 1 and 500 that are not divisible by any one of the integers 2,3 and 5.
15.a) Solve a n – 6a n-1 + 8a n-2 = 3n.
b) Using mathematical induction